Krull-Tropical Hypersurfaces

Fuensanta Aroca[1]

  • [1] Instituto de Matemáticas (Unidad Cuernavaca) Universidad Nacional Autónoma de México. A.P. 273, Admon. de correos #3 C.P. 62251 Cuernavaca, Morelos

Annales de la faculté des sciences de Toulouse Mathématiques (2010)

  • Volume: 19, Issue: 3-4, page 525-538
  • ISSN: 0240-2963

Abstract

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The concepts of tropical semiring and tropical hypersurface, are extended to the case of an arbitrary ordered group. Then, we define the tropicalization of a polynomial with coefficients in a Krull-valued field. After a close study of the properties of the operator “tropicalization" we conclude with an extension of Kapranov’s theorem to algebraically closed fields together with a valuation over an ordered group.

How to cite

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Aroca, Fuensanta. "Krull-Tropical Hypersurfaces." Annales de la faculté des sciences de Toulouse Mathématiques 19.3-4 (2010): 525-538. <http://eudml.org/doc/115858>.

@article{Aroca2010,
abstract = {The concepts of tropical semiring and tropical hypersurface, are extended to the case of an arbitrary ordered group. Then, we define the tropicalization of a polynomial with coefficients in a Krull-valued field. After a close study of the properties of the operator “tropicalization" we conclude with an extension of Kapranov’s theorem to algebraically closed fields together with a valuation over an ordered group.},
affiliation = {Instituto de Matemáticas (Unidad Cuernavaca) Universidad Nacional Autónoma de México. A.P. 273, Admon. de correos #3 C.P. 62251 Cuernavaca, Morelos},
author = {Aroca, Fuensanta},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {tropical hypersurfaces; Kapranov's theorem; valuations},
language = {eng},
number = {3-4},
pages = {525-538},
publisher = {Université Paul Sabatier, Toulouse},
title = {Krull-Tropical Hypersurfaces},
url = {http://eudml.org/doc/115858},
volume = {19},
year = {2010},
}

TY - JOUR
AU - Aroca, Fuensanta
TI - Krull-Tropical Hypersurfaces
JO - Annales de la faculté des sciences de Toulouse Mathématiques
PY - 2010
PB - Université Paul Sabatier, Toulouse
VL - 19
IS - 3-4
SP - 525
EP - 538
AB - The concepts of tropical semiring and tropical hypersurface, are extended to the case of an arbitrary ordered group. Then, we define the tropicalization of a polynomial with coefficients in a Krull-valued field. After a close study of the properties of the operator “tropicalization" we conclude with an extension of Kapranov’s theorem to algebraically closed fields together with a valuation over an ordered group.
LA - eng
KW - tropical hypersurfaces; Kapranov's theorem; valuations
UR - http://eudml.org/doc/115858
ER -

References

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  2. Einsiedler (M.), Kapranov (M.), and Lind (D.).— Non-Archimedean amoebas and tropical varieties. J. Reine Angew. Math., 601:139-157 (2006). arXiv:math.AG/0408311v2. Zbl1115.14051MR2289207
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  5. Itenberg (I.), Mikhalkin (G.), and Shustin (E.).— Tropical algebraic geometry, volume 35 of Oberwolfach Seminars. Birkhäuser Verlag, Basel (2007). Zbl1162.14300MR2292729
  6. Katz (E.).— A tropical toolkit. Expositiones Mathematicae, 27(1):1-36 (2009). arXiv:math/0610878. Zbl1193.14004MR2503041
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  8. Ribenboim (P.).— The theory of classical valuations. Springer Monographs in Mathematics. Springer-Verlag, New York (1999). Zbl0957.12005MR1677964
  9. Richter-Gebert (J.), Sturmfels (B.), and Theobald (T.).— First steps in tropical geometry. In Idempotent mathematics and mathematical physics, volume 377 of Contemp. Math., pages 289-317. Amer. Math. Soc., Providence, RI, (2005). arXiv:math.AG/0306366. Zbl1093.14080MR2149011
  10. Shafarevich (I. R.).— Basic algebraic geometry. 1. Springer-Verlag, Berlin, second edition, (1994). Varieties in projective space, Translated from the 1988 Russian edition and with notes by Miles Reid. Zbl0362.14001MR1328833
  11. Spivakovsky (M.).— Valuations in function fields of surfaces. Amer. J. Math., 112(1):107-156 (1990). Zbl0716.13003MR1037606
  12. Zariski (O.) and Samuel (P.).— Commutative algebra. Vol. II. Springer-Verlag, New York, (1975). Reprint of the 1960 edition, Graduate Texts in Mathematics, Vol. 29. Zbl0313.13001MR389876

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